22 June 2008

Nonconvergent sums and evolution

It's often said that evolution works in such a way as to maximize the number of descendants that an individual has.

More formally, X's children share one-half of their genes with X, X's grandchildren share one-quarter of their genes with X, and more generally X's nth-generation descendants share 1/2n of their genes with X. So if you buy the whole "selfish gene" theory that genes act in such a way as to maximize the number of copies of them which are made, the quantity individuals should be attempting to maximize might be half the number of children, plus one fourth the number of grandchildren, plus one eighth the number of great-grandchildren, and so on.

It's an infinite sum.

What's more, if you assume a "total fertility rate" of k -- that is, the average female bears two children -- then this sum is k/2 + k2/4 + k3/8 + .... And if k = 2, which corresponds to population not growing or shrinking, this sum is just 1 + 1 + 1 + 1 + ... which doesn't converge. (Similarly if k > 2, but populations which grow indefinitely don't seem sustainable.)

Of course, biologically populations don't live for infinitely long. And in reality people don't at least consciously think more than a couple generations into the future. So practically speaking this is all a bit meaningless.

edited, Monday, 8:42 am: I did mean maximizing the number of copies of genes, in the sense of Dawkins' idea of the selfish gene, and this post is not meant to be taken seriously.

14 comments:

"It's often said that evolution works in such a way as to maximize the number of descendants that an individual has."

Huh? What? Often said by whom?

Clearly evolution is a process whereby the number of descendants of some individuals is also minimized.

Natural selection is a process whereby inheritance of helpful traits is increased (maximized in some sense? I'm not sure, but perhaps), and harmful traits are decreased. This selects against some individuals, of course.

I think it's really, really mistaken to conceive of this in terms of maximizing an individual's descendants.

Evolution, working at the level of genes, maximizes the presence of those genes that contribute to fitness. It also maximizes the fitness of particular species, which presumably is related to maximizing the descendants of those individuals with particularly advantageous genes, but this is quite a different thing from what you said.

The maximizing behavior, as I understand it, is a characterization by Dawkins and is not in terms of descendents, but in terms of surviving copies of genes. It is a metaphor and even if you are to take the expected behavior given that characterization seriously I think there is a "tends to" somewhere.

As far as I know, from the scientists I know or whose writings I read, that "evolution" works to "maximize descendants" is a pretty severe misunderstanding... though probably one of the prevailing pop-misconceptions?

Don't take me too seriously, but I'd encourage you to double-check for your own sake.

Last year I did a careful reading of The Selfish Gene and The Extended Phenotype and exactly this problem struck me: what exactly is Dawkins' mathematical model and is it coherent?

Here's from a blog entry I made at the time: "The difficulty that I found in Dawkins is this: he ascribes two different goals to his 'selfish genes'. Most of the time they are said to be maximizing 'survival' (The Extended Phenotype, p. 233) but sometimes they are described as striving to 'increase' (Phenotype, p. 84). When he turns from theoretical statements to examples, Dawkins seems to reverse this preference. The genes in his examples usually strive to increase the number of 'germline' genes. (Germline genes are those actually in reproductive lineages.)"

I worked the implications of the goals (survival vs. increase), and, for the reason you suggest and others, feel that I showed that an "increase" goal is incoherent as a mathematical model, while a "survival" goal is not.

This is not a philosophical issue. The difference is probably testable -- the two goals certainly result in models which make different predictions.

To see that there is a difference in mathematical models, it might help to consider the difference between investing for growth and investing for preservation of capital. The two are usually not the same, and it usually requires different investment vehicles to deliver each one .

Isabel, can you explain why it is necessary that the derivative of the area of a circle is its circumference and likewise surface area for a sphere. I was in my (remedial) calculus class the other day and the instructor wouldn't listen when I pointed this out and, when I said it made sense that the integral of all the circumferences of all the circles from 0 to r would be the area of the circle... then he started muttering something about polar notation which i didn't understand and I was at the limit of my ability to argue my point further since I don't actually know any maths.

To see why the circumference of a circle of radius r (2πr) is the derivative of its area (π r^2), let's pretend that we know A(r) = π r^2 is the area of a circle of radius r, but we don't know what the circumference is, but let's call it C(r). How would we find the circumference?

So we begin by considering two concentric circles. Let one of these circles have radius r and the other have radius r + dr, where "dr" some small length. The difference between the areas of the two circles is nearly A'(r) dr. But the difference is just the area of a ring (or "annulus") with inner radius r and outer radius r + dr. This ring goes around the circle, and you can imagine "unrolling" this ring into a rectangle with sides dr and C(r). Thus it has area C(r) dr, and so A'(r) dr = C(r) dr, so A'(r) = C(r).

It's not really formally correct to use an equals sign here, though, because these quantities aren't actually equal, and what does "dr" mean anyway? But this explains why the derivative of the area of a circle is the circumference. The same thing is true for the volume and surface area of a sphere; you get a hollow sphere instead of an annulus but the idea is the same.

Now, you might ask why this doesn't seem to work for, say, the surface area of a cube. The volume of a cube of side s is s^3, its surface area is 6s^2. But that's because the side length of a cube is analogous to the diameter, not the radius. If we let s = 2r, so r is half the side length and thus analogous to the radius, then the volume is 8r^3 and the surface area 24r^2.

...and what does "dr" mean anyway? Why, it's just the other standard coordinate on the tangent bundle, in addition to r. V.I. Arnold at the beginning of section 3 of chapter 1 of his Geometrical Methods in the Theory of Ordinary Differential Equations writes: "...dx, dy, dp are not some mystical infinitely small quantities, but rather particular linear functions of the tangent vector ξ."

I guess this doesn't matter since A'(r) dr != C(r) dr but if you "unrolled" the annulus wouldn't you get a trapazoid of height dr with bases = C(r) and C(r+dr) so then can't you not set A'(r) dr = C(r).

Uh again, I don't know maths, but i'm trying to learn after a misguided youth, so your indulgence is appreciated...

you're right that the bases of the unrolled annulus are actually of different lengths, but r is very close to r + dr (the idea of the analysis being that dr is very small compared to r), and C is a continuous function, so C(r) and C(r+dr) are very close to each other -- close enough that for our purposes we'll treat them as being equal.

(Of course, one can't always treat things that are close to each other in this way.)

OK so how about this. So i'm pretty sure that annulus stretched out is an isosceles trapezoid, though i don't know how (and do not feel trying to figure out) to prove that. So the area is h(b1+b2)/2. where b1 is C(r) and b2 is C(r+dr) and h is dr. so A(r)=(dr(2*Pi*r+2*Pi*(r+dr))/2 or A (r) = 2*dr*Pi*r+dr^2*pi. So as r approaches zero, the area of the trapezoid approaches the area of of the circle with radius dr+r. So as dr approaches zero the "area" it seems like of the trapezoid should approach 2*Pi*r (like in the instant before goes to zero), but maybe this is just wishful thinking and the result of me using circumfrences for b1 and b2.

ok, well i have to go back to calculus now... i wonder if any of that made sense....

yeah so i started typing that before you typed your comment about analysis, which i am only aware of in so much as i've seen it written in the course catalog. can i use this "analysis" thing to address my issue of "as dr approaches zero the "area" it seems like of the trapezoid should approach 2*Pi*r (like in the instant before goes to zero)" or am i just making stuff up now?

Ok so it just occurred to me that i can just undistribute the dr so that A(r)/dr=2*pi*r+dr*2*pi (duh) so as dr -> 0, A(r) = 2*pi*r. which is legal, i guess since that's how differentiation works (which is what i think i just stumble into doing backwards here anyway). though it occurs to me that as the denominator approaches zero the quotient should go to infinity which makes me question how differentiation is possible. except that the quantity in the denominator never goes quite to zero, but then that raises the question of how we can assume away the terms in the numerator that are the denominator raised to some power (in the fundamental equation of calculus). And i'm thinking this all has something to do with that analysis stuff again.

But anyway, A(r)/dr=2*pi*r+dr*2*pi as dr -> 0, A(r) = 2*pi*r. and A (r) = 2*dr*Pi*r+dr^2*pi as r -> 0, A(r) = pi*dr^2. so doesn't that "prove" that the rate of change of the area is and must necessarily be the circumference?